_B._ The difference is manifest. For when you multiply a number into lines, the product is lines; as the number 2 multiplied into 3 lines is no more than 3 lines 2 times told. But if you multiply lines into lines you make planes, and if you multiply lines into planes you make solid bodies. In geometry there are but three dimensions, lengths, superficies, and body. In arithmetic there is but one, and that is number or length which you will. And though there be some numbers called plane, other solids, others plano-solid, others square, others cubic, others square-square, others quadrato-cubic, others cubi-cubic, &c., yet are all these but one dimension, namely number, or a file of things numbered. _A._ But seeing this way of calculation by numbers is so apparently false, what is the reason this calculation came so near the truth? _B._ It is because in arithmetic units are not nothings, and therefore have breadth. And therefore many lines set together make a superficies though their breadth be insensible. And the greater the number is into which you divide your line, the less sensible will be your error. _A._ Archimedes, to find a straight line equal to the circumference of a circle, used this way of extracting roots. And it is the way also by which the table of sines, secants, and tangents have been calculated. Are they all out? _B._ As for Archimedes, there is no man that does more admire him than I do: but there is no man that cannot err. His reasoning is good. But he, as all other geometricians before and after him, have had two principles that cross one another when they are applied to one and the same science. One is, that a point is no part of a line, which is true in geometry, where a part of a line when it is called a point, is not reckoned; another is, that a unit is part of a number; which is also true; but when they reckon by arithmetic in geometry, there a unit is sometimes part of a line, sometimes a part of a square, and sometimes part of a cube. As for the table of sines, secants, and tangents, I am not the first that find fault with them. Yet I deny not but they are true enough for the reckoning of acres in a map of land. _A._ What a deal of labour has been lost by them that being professors of geometry have read nothing else to their auditors but such stuff as this you have here seen. And some of them have written great books of it in strange characters, such as in troublesome times, a man would suspect to be a cypher. _B._ I think you have seen enough to satisfy you, that what I have written heretofore concerning the quadrature of the circle, and of other figures made in imitation of the parabola, has not been yet confuted.